Intuitively, this choice implies that the cost of transporting one
unit of mass from a source point \({\bf x} \in {\mathcal {X}}\)  to a
target point \({\bf y} \in {\mathcal {Y}}\)  corresponds to the distance
between the two points.
The optimal cost \(T_C({\bf f}, {\bf g})\)  in (REF )
corresponding to the cost matrix (REF ) induces a metric known as the Wasserstein
distance of order \(p\)  [1]}, [2]},
\(W_p({\bf f}, {\bf g}) = \bigl ( T_{C}({\bf f}, {\bf g}) \bigr )^{1/p}.\) 
